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ECE 595, Section 10Numerical Simulations

Lecture 3: Computability

Prof. Peter Bermel

January 11, 2013

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Outline

Overview

Definitions

Computing Machines Church-Turing Thesis

Polynomial Time

Example

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Overview from First Lecture

Study of the complexity of

algorithms

Based on Turing machines

Often, one compares

algorithms for best scaling

in large problems

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Alan Turing (from University

of Calgary Centenary event)

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Definitions

Algorithm set of rules for computing a functionmapping input data to output

Specific realizations of computers: Turing machines

Lambda calculus

-recursive functions

Register machines

Running time number of basic operationsrequired

Efficient computability

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Turing Machine

Performs the following

elementary operations:

Read a bit of input and byte of

data from work

Write a bit to work

Choose next state Q. Either:

Stop and output a 0 or 1

Choose a new rule to apply next

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Turing Machine: Behavior in Each State

qstart: write start symbol, change state to qcopy

qcopy: copy non-blank symbol, otherwise go to

qright

qright: go back to start, then qtest

qtest: if at start and work is blank, write 1;

else, write 0 qhalt: finished

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Time Constructible Functions

A function T: N N is time constructible iff

T(n) n

TM M computes x T(x) in time T(n)

Examples: n, n log n, 2n

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Turing Machine: Key Properties

If f is computable in T(n).

with alphabet , it is computable in 4 log T(n)

with TM

with k tapes, its computable in 5kT(n)2 with TM

with bidirectional tape, computable in 4T(n) with

TM

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Universal Turing Machines

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Other Computing Machines

Register machines

Post-Turing machine: adds jump and erasefunctions

Many other variations, getting closer to assemblylanguage

Lambda calculus basis for Scheme and LISP

-recursive functions partial functionsmapping NN

Which one should we use, in which situations?

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Church-Turing Thesis

Every physically realizable computing machine

can be simulated by a Turing machine, or any

other type of machine

This thesis has not been rigorously proven

Not applicable for non-deterministic systems

Quantum computers have caused some to re-

evaluate CT thesis

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Uncomputability

It may seem obvious everything is computable

with enough time, but

Theres always an uncomputable function UC

Another example: the halting problem

Computes whether another function will finish

Cannot be computed in general

Also: standard maximum productivity function

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Big-Oh Notation

Take two functions f,g mapping NN, there

exists a constant c such that f(n)cg(n)

Sometimes write f(n) = O(g(n))

Examples

If f(n)=100n2+24n+2logn and g(n)=n2, then f=O(g)

If f(n)=2n and g(n)=nc for every c in N, then g=O(f)

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Polynomial Complexity

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Other Classes

Turing machines with randomness class BPP

Quantum computers class BQP

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Examples

CONNECTED the set of all connected graphs

TRIANGLEFREE the set of all graphs without

a triangle

BIPARTITE the set of all bipartite (distinct)

graphs

TREE the set of all trees

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Next Class

Discussion of computational complexity Please read Chapter 2 of Computational

Complexity: A Modern Approach by Arora& Barak

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http://www.cs.princeton.edu/theory/complexity/NPchap.pdfhttp://www.cs.princeton.edu/theory/complexity/NPchap.pdfhttp://www.cs.princeton.edu/theory/complexity/NPchap.pdfhttp://www.cs.princeton.edu/theory/complexity/NPchap.pdfhttp://www.cs.princeton.edu/theory/complexity/NPchap.pdfhttp://www.cs.princeton.edu/theory/complexity/NPchap.pdfhttp://www.cs.princeton.edu/theory/complexity/NPchap.pdf